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Primitive sequence number
Primitive sequence number is the large number made by BashicuUser:BashicuHyudora. It was posted to the large number thread Part.10Although 2channel is banned in China and it bans the access from United States, the archive can be seen here. of website 2channel as the program (message No.109) and its examples (No.135-140)the article explaining itGoogology in Japan - exploring large numbers. The algorithm making primitive sequence number is called as Primitive sequence system, it is like Beklemishev's worm and it has the strength of \(f_{\epsilon_0}(n)\). The size of primitive sequence number is about \(f_{\epsilon_0+1}(10)\). Pair sequence system, which is the expansion of primitive sequence system into two rows, has the strength of \(f_{\psi(\Omega_{\omega})}(n)\). The expansion into three rows is called as Trio sequence system. The generalization of them are called as Bashicu matrix system. It has \(n\) rows. Their investigation is currently in progress. Definition The first definition of the primitive sequence number was posted into the communication website 2channel large number thread Part.10its archive as the pseudo code in BASIC language. After that, the definitions are maintained on the article Summary of the large number in BASIC Language in Wikia user blog. Mathematical definition Although the original definition is written in the pseudo code of BASIC language, It can be described in the mathematical definition as following: Primitive sequence is a list of non negative integer \(S = (S_0, S_1, \ldots, S_k)\). Primitive sequence has the behavior as the function from a non negative integer \(n\) into a non negative integer, the value of \(Sn\) is defined as below: * \( ()n=n \). * good part \(g\) and bad part \(b\) of the primitive sequence are defined as following: (\(i\) is the largest non negative integer which gives \(rby Fish and by koteitan The inverse correspondence from ordinals to primitive sequences of standard form is demonstrated in the following way: # Write the hydra tree corresponding to a given ordinal \(\alpha\). The figure above represents \(\omega^{\omega^\omega+\omega^3}+\omega^{\omega^2+1}\) as the paper by Kirby and Paris (1982).Kirby, L.; Paris, J. (1982), "Accessible independence results for Peano arithmetic" # Start below from root node. # Add a new element 0 into the last of the sequence when you go up in 1 above from the root node. # Add a new element "(the last element) + 1" into the last of the sequence when you go up in 1 above from the other node. # When you reach the end of the branch and go down to the branching node (common anscestor) and go on the another branch, add "(the branching element) + 1". # so that the number in the node becomes the value "(height)-1". # Here is the example to corresponding with the figure like this. The part of \(\omega^{\omega^\omega}\) is shown as (0,1,2,3). after that, go down 3 nodes, go to the next branch (segment), get (1,2,2,2), it is added into the sequence. After that, (0,1,2,2,1) is added as the same. Finally, the hydra tree is corresponded withthe primitive sequence (0,1,2,3,1,2,2,2,0,1,2,2,1). #\(\omega^{\omega^\omega+\omega^3}+\omega^{\omega^2+1} = (0,1,2,3,1,2,2,2,0,1,2,2,1)\). Examples Here are examples of the correspondence between ordinals below \(\varepsilon_0\) and primitive sequences of standard form. \begin{eqnarray*} 1 &=& (0) \\ 2 &=& (0,0) \\ 3 &=& (0,0,0) \\ \omega &=& (0,1) \\ \omega+2 &=& (0,1,0,0) \\ \omega \cdot 2 &=& (0,1,0,1) \\ \omega^2 &=& (0,1,1) \\ \omega^2+\omega &=& (0,1,1,0,1) \\ \omega^3 &=& (0,1,1,1) \\ \omega^\omega &=& (0,1,2) \\ \omega^{\omega^\omega} &=& (0,1,2,3) \\ \omega^{\omega^{\omega^\omega}} &=& (0,1,2,3,4) \\ \omega^{\omega^{(\omega^\omega+1)}} &=& (0,1,2,3,4,2) \\ \omega^{\omega^{\omega^{\omega^\omega}}} &=& (0,1,2,3,4,5) \\ \end{eqnarray*} Programs The shortest codes The python code which performs Primitive sequence system is shown in the site [https://docs.google.com/document/d/1Pym015m3fLEL25mE_qyWkZXp1HhwGoPPp6aX_tyAPKY/edit The Py_1 Function], which is the online JAM to make the large number with short codes. The code is 72 bytes and it calculates (0)(1)(2)(3)(4)(5)(6)(7)(8)(9)9 in \(f(n)=n^2\). x,*m=range(9,-1,-1) while m: q,*m=m;x*=x if q:m=m:m.index(q-1)+1*x+m Here are a few codes: * Expansion calculators: ** in C: Bashicu matrix calculator ** in Javascript: yaBMS * Ordinal converters: ** in C: primseq ** in Javascript: p2o * Hydra notation converters: ** in Javascript: hydraviewer(for Pair sequence system) ** in Javascript: BMS Hydra viewer(Multi-row supported) References See also zh:原始數列系統 ja:原始数列数 Category:Bashicu matrix system Category:Numbers Category:Googology in Asia